In-orbit capture of space objects is a crucial aspect of spacecraft servicing, whether it involves space debris removal or spacecraft maintenance. The challenging task of capturing objects in space is the primary focus of this article. This paper provides a comprehensive review of the dynamics and control issues related to space robot capture. It begins by introducing major space robot programs both domestically and internationally. Subsequently, it delves into the dynamics and control problems during the pre-capture, mid-capture, and post-capture phases. It is hoped that the content of this article will be beneficial to scholars engaged in space robot technology research.